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A-Level CAIE Mathematics A21.5 TrigonometryQuestion Bank

1.5 Trigonometry

Question 1

[Maximum number: 4]

Given that θ\theta is an obtuse angle measured in radians and that sin⁡θ=k\sin \theta=k, find, in terms of k, an expression for

Question 1(i)

(a)

cos⁡θ\cos \theta,

[ 1 ]

Question 1(ii)

(b)

tan⁡θ\tan \theta,

[ 2 ]

Question 1(iii)

(c)

sin⁡(θ+π)\sin (\theta+\pi).

[ 1 ]

Question 1

[Maximum number: 2]
Question image

The diagram shows part of the graph of y=a+bsin⁡xy=a+b \sin x. State the values of the constants a and b.

Question 1

[Maximum number: 3]

Solve the equation 4sin⁡θ+tan⁡θ=04 \sin \theta+\tan \theta=0 for 0∘<θ<180∘0^{\circ}<\theta<180^{\circ}.

Question 1

[Maximum number: 3]

Given that cos⁡x=p\cos x=p, where x is an acute angle in degrees, find, in terms of p,

Question 1(i)

(a)

sin⁡x\sin x,

[ 1 ]

Question 1(ii)

(b)

tan⁡x\tan x,

[ 1 ]

Question 1(iii)

(c)

tan⁡(90∘−x)\tan \left(90^{\circ}-x\right).

[ 1 ]

Question 1

[Maximum number: 3]

Solve the equation 8sin⁡2θ+6cos⁡θ+1=08 \sin ^{2} \theta+6 \cos \theta+1=0 for 0∘<θ<180∘0^{\circ}<\theta<180^{\circ}.

Question 1

Question 1(i)

(a)

Prove the identity tan⁡2θ−sin⁡2θ≡tan⁡2θsin⁡2θ\tan ^{2} \theta-\sin ^{2} \theta \equiv \tan ^{2} \theta \sin ^{2} \theta.

[ 3 ]

Question 1(ii)

(b)

Use this result to explain why tan⁡θ>sin⁡θ\tan \theta>\sin \theta for 0∘<θ<90∘0^{\circ}<\theta<90^{\circ}.

[ 1 ]

Question 1

[Maximum number: 4]

The acute angle x radians is such that tan⁡x=k\tan x=k, where k is a positive constant. Express, in terms of k,

Question 1(i)

(a)

tan⁡(π−x)\tan (\pi-x),

[ 1 ]

Question 1(ii)

(b)

tan⁡(12π−x)\tan \left(\frac{1}{2} \pi-x\right),

[ 1 ]

Question 1(iii)

(c)

sin⁡x\sin x.

[ 2 ]

Question 1

[Maximum number: 5]

Solve the equation

sec⁡2θ+5tan⁡2θ=9+17sec⁡θ\sec ^{2} \theta+5 \tan ^{2} \theta=9+17 \sec \theta

for 0∘<θ<360∘0^{\circ}<\theta<360^{\circ}.

Question 1

Question 1(i)

(a)

Show that the equation

3(2sin⁡x−cos⁡x)=2(sin⁡x−3cos⁡x)3(2 \sin x-\cos x)=2(\sin x-3 \cos x)

can be written in the form tan⁡x=−34\tan x=-\frac{3}{4}.

[ 2 ]

Question 1(ii)

(b)

Solve the equation 3(2sin⁡x−cos⁡x)=2(sin⁡x−3cos⁡x)3(2 \sin x-\cos x)=2(\sin x-3 \cos x), for 0∘⩽x⩽360∘0^{\circ} \leqslant x \leqslant 360^{\circ}.

[ 2 ]

Question 1

[Maximum number: 4]

Solve the equation 2cos⁡θ=7−3cos⁡θ2 \cos \theta=7-\frac{3}{\cos \theta} for −90∘<θ<90∘-90^{\circ}<\theta<90^{\circ}.

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